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NEW YORK — Home prices tumbled by the steepest rate ever in May, according to a closely watched housing index released Tuesday, as the housing slump deepened nationwide.

The Standard & Poor’s/Case-Shiller 20-city index dropped by 15.8 percent in May compared with a year ago, a record decline since its inception in 2000. The 10-city index plunged 16.9 percent, its biggest decline in its 21-year history.

No city in the Case-Shiller 20-city index saw price gains in May, the second straight month that’s happened. The monthly indices have not recorded an overall home price increase in any month since August 2006.

Home values have fallen 18.4 percent since the 20-city index’s peak in July 2006.

Nine metropolitan cities — Las Vegas, Miami, Phoenix, Los Angeles, San Diego, San Francisco, Seattle, Wash., Portland, Ore., and Washington, D.C. — posted record lows in May. And the value of housing in Detroit is now lower than it was in 2000.